A Central Limit Theorem for Random Walks on the Dual of a Compact Grassmannian

Margit Rösler ; Michael Voit

Symmetry, integrability and geometry: methods and applications, Tome 11 (2015)

Résumé

We consider compact Grassmann manifolds $G/K$ over the real, complex or quaternionic numbers whose spherical functions are Heckman–Opdam polynomials of type $BC$. From an explicit integral representation of these polynomials we deduce a sharp Mehler–Heine formula, that is an approximation of the Heckman–Opdam polynomials in terms of Bessel functions, with a precise estimate on the error term. This result is used to derive a central limit theorem for random walks on the semi-lattice parametrizing the dual of $G/K$, which are constructed by successive decompositions of tensor powers of spherical representations of $G$. The limit is the distribution of a Laguerre ensemble in random matrix theory. Most results of this paper are established for a larger continuous set of multiplicity parameters beyond the group cases.

Détail

Keywords: Mehler–Heine formula; Heckman–Opdam polynomials; Grassmann manifolds; spherical functions; central limit theorem; asymptotic representation theory.

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